Fibonacci Calculator — nth number and sum

How it works

The Fibonacci sequence is built by one simple rule: start with two 1s, and every term after that is the sum of the two before it:

Fₙ = Fₙ₋₁ + Fₙ₋₂, with F₁ = F₂ = 1

That gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … — each step just adds the previous two. To get the nth number you do not need a special formula; you can build up to it by repeated addition, which is exactly what this calculator does.

The sum of the first n Fibonacci numbers has a tidy shortcut. Adding them one by one always lands on the term two places further along, minus one:

F₁ + F₂ + … + Fₙ = Fₙ₊₂ − 1

The sequence also hides the golden ratio: divide any term by the one before it and the result closes in on φ ≈ 1.618 as the numbers grow — 55 ÷ 34 ≈ 1.618. (This tool caps n at 78, the largest Fibonacci number that stays exact in standard computer arithmetic.)

Worked example

For n = 10, count along the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 — the 10th term is 55. Their sum is 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143, which the shortcut gives instantly as F₁₂ − 1 = 144 − 1 = 143.

Frequently asked questions

What is the Fibonacci sequence?

It is the sequence 1, 1, 2, 3, 5, 8, 13, 21, … in which each number is the sum of the two before it. It is named after Leonardo of Pisa ("Fibonacci") and appears throughout mathematics and nature, from flower petals to spiral shells.

What is the nth Fibonacci number?

It is the number in position n. Because Fₙ = Fₙ₋₁ + Fₙ₋₂, you find it by adding your way up the sequence: F₇ = 13, F₁₀ = 55, F₂₀ = 6,765. Enter n above and the calculator does the adding for you.

How do I find the sum of Fibonacci numbers?

Use the shortcut: the sum of the first n terms equals Fₙ₊₂ − 1. For the first ten terms, that is F₁₂ − 1 = 144 − 1 = 143, the same as adding them one by one.

How is Fibonacci related to the golden ratio?

The ratio of each term to the one before it approaches the golden ratio φ ≈ 1.618 as the terms grow: 8 ÷ 5 = 1.6, 55 ÷ 34 ≈ 1.618, 6,765 ÷ 4,181 ≈ 1.61803. The bigger the numbers, the closer the ratio.

Does the sequence start with 0 or 1?

Both conventions exist. This calculator uses F₁ = F₂ = 1, so the sequence begins 1, 1, 2, 3, … Some texts start at F₀ = 0, giving 0, 1, 1, 2, 3, … — the numbers are the same, just shifted by one index.

How large can a Fibonacci number get?

They grow exponentially — roughly by a factor of the golden ratio each step. This calculator stops at the 78th term (about 8.9 quadrillion), the largest that stays an exact whole number in standard computer arithmetic.